Briefly, no. Words mean what people use them to mean, and words can have multiple meanings. Expressions like "My philosophy is..." are so common that they represent one of the meanings that the word "philosophy" has come to have in English. Of course it's not what professional philosophers usually have in mind when they use the word, but professional philosophers don't get veto power over usage.
A further thought or two: I can't really get bent very far out of shape by this one, but it's a bit unfortunate in at least one respect: it gives some people the impression that philosophy is all about truisms and banal principles instead of being something that calls for rigorous thought. But the cure for that is not to complain about how folks use the word; it's for philosophers to do a better job of helping people understand what we do and why it's worth doing.

Think about water. When a lexicographer asks what the word "water" means, s/he is asking how the word is used. It's an empirical question about people's actual linguistic behavior. If it turned out that enough people use the word "water" to refer to vodka, then "vodka" would be one of the meanings of the word "water." But when a chemist answers the question "What is water?", s/he's not telling us how the word is used. On the contrary, s/he will assume that that's settled. The question isn't about the word; it's about the stuff that we use the word to refer to. What is it? And, at least close enough for present purposes, the substance that we refer to by using the word "water" is the liquid composed (mostly) of molecules of H2O.
The chemist's answer can be quite different from the lexicographer's. A lexicographer could have come up with a perfectly acceptable account of the meaning-in-use of the word "water" before we knew what water is. Many people who use the word "water" correctly don't know...

You write that one response of professional philosophers is "they don't let the doubts affect their lives, which is contrary to what an intellectually capable human being (i suppose philosophers are of course) should do "
Here's my question: why is this what an "intellectually capable human being" should do? I don't think that's even remotely obvious.
There's more or less nothing that I couldn't bring myself to doubt if I really tried. Maybe 1+2 = 3 is an exception. On the other hand, 5879+3627=9506 is something I likely could get myself to feel unsure about with a lot of effort.
But so what? It's consistent to doubt any empirical proposition. And it's psychologically possible to doubt a lot of non-empirical propositions by worrying about whether one's cognitive engine has gone on the fritz. But while all of this is true, it seems to me to have no force. The best way I've ever seen anyone make the point is the way my one-time colleague Dudley Shapere used to put it:
The possibility...

If by "assume," you mean "accept without argument," then the answer is yes to the first part of your question and no to the second.
Yes: philosophers assume all sorts of things. They usually assume that there is a world out there and that there are people who at least potentially can read and respond to their arguments. They very often take for granted all sorts of facts, scientific and garden-variety: that there are trees; that people sometimes do things deliberately; that water is made of H2O; that the 4-color conjecture has been successfully proven.
No: this isn't contrary to the intent of philosophy. Except on very eccentric views, philosophy is not the enterprise of doubting everything that can be doubted and accepting only what can be proved from indubitable premises. That may have been Descartes' project, but it's been almost no one else's. In fact, most philosophers would say that this project is deeply flawed.
Philosophy, like the Odyssey begins in the middle of things. Want to think...

A not-really-relevant aside: most philosophers don't own philosophical dictionaries, but let that pass.
Here's a non-philosophical word: unicorn. It's easy to explain what it means, but there aren't any real-life examples. Whether a term can be clearly explained and whether there are actual examples are quite different questions.
If it's controversial whether something exists then a philosopher shouldn't pretend otherwise, but the question of whether something exists (God, for instance) might be an interesting one. And sometimes the best way to understand a notion is a philosophical question. The notion of free will is like that. In that case, there's no one meaning for the term, but it's possible to have a perfectly reasonable discussion of what might count as free will and why some answers might be better than others. In fact that sort of discussion comes up in many disciplines. Philosophers are unusual in that they're trained to notice this kind of unclarity and to reason about it carefully.

I'm not entirely sure I follow, but perhaps this will be of some use. Whether two propositions really have four possible combinations of truth values depends on the propositions. Non-philosophical examples make the point easier to follow. Suppose P is "Paula is Canadian" and Q is "Quincy is Australian." In this case, the two propositions are logically independent, and all four combinations P&Q, P&~Q, ~P&Q and ~P&~Q represent genuine possibilities. But not all propositions are independent in this way; it depends on their content. P and Q might be contradictories, that is, one might be the denial of the other. (If P means that Paula is Canadian and Q means that she is not Canadian, then we have this situation.) In that case, the only two possibilities are P&~Q and ~P&Q. Or P and Q might be contraries, meaning that they can't both be true though they could both be false. For example: if P is "Paula is over 6 feet tall" and Q is "Paula is under 5 feet tall," then we only have three...

Not sure I follow, but by "concrete" I'm guessing you mean either "objective" or "easy to settle." If you do, then on either alternative I can't see why there would be any difference between the two. In any case, the way you've put things suggests that nose-counting may be relevant. That's surely wrong. The percentage of philosophers who think Israel does or doesn't have a right to exist doesn't seem to me to tell us much of anything about whether that's the best view of the matter; likewise for questions about epistemology. What really matters are the reasons. I'd add this, however: most philosophers have spent a fair bit of time thinking about epistemological questions; it's part of their training. And so if most philosophers held a particular epistemological view, that would be interesting and might suggest something about the weight of the arguments. However, most philosophers have not spent much of their training thinking about political philosophy; insofar as we can talk about expertise here,...

To say that all questions demand a philosophical response (whatever exactly that is) would be at best a very controversial philosophical view. And a philosopher who thought that philosophers should have more say on large practical questions than anyone else would be hard pressed to justify his or her position. To take your example, the question of whether there should be a united Ireland has many parts. Some of those parts no doubt call for philosophical reflection but some don't. (For example: what people in Northern Ireland and in the Republic actually think is surely relevant; but isn't something we can sort out by doing philosophy.) And even the philosophical aspects (having to do, say, with how we balance competing values) needn't be addressed by professional philosophers; philosophers don't have a monopoly on philosophical thinking. Of course, there's a more straightforward way to deal with questions of the form "Do philosophers think X?" If X is something controversial (and often even if it...

I'd like to pause over the first half of your first sentence (the 'if' bit): the idea that philosophy is difficult to understand and difficult to put into practice. I'd suggest that this isn't the best way to put things. Philosophy may or may not be difficult to understand, but no more so than any number of other subjects. Philosophy can be difficult to follow because when done well, it depends on careful arguments and subtle distinctions. That means there's a lot to keep track of, even if the writing is crystal clear. Compare: each step in a math proof might be clear by itself; seeing the argument entire might not be easy. The next bit is supposed to be that philosophy is difficult to "put into practice." What's striking here is that very few of the philosophers I know think of philosophy as something you "put into practice" in the way that, for example, I might put the Golden Rule into practice. By and large, philosophy isn't in the business of giving practical advice. For example: some...

I'm guessing that by "personal philosophical beliefs" you mean not just philosophical beliefs that someone might happen to hold (such as the belief that numbers exist as Platonic objects, for example) but beliefs and commitments about matters that someone takes to be of great personal significance—the kinds of things one might build one's life around. To clarify: I might think that numbers really exist as abstract objects, but if I were talked out of that belief, not much about how I live my life would change. For many religious people, on the other hand, religious beliefs are part of their core. A committed Christian, for example, might well think that if she lost her Christianity, there's an important sense in which she wouldn't be the same person. However, as you apparently recognize, religious beliefs aren't the only ones in that category. Ethical commitments are a good example. Many people with deep ethical commitments are not religious, and yet those commitments are every bt as important to...